The stronger $s$-Cohen–Macaulay $M$-vector conjecture

Let $s$ be a positive integer, and let $\Delta$ be an $s$-Cohen–Macaulay complex of dimension $d-1$, with $h$-vector $(h_0,h_1,\ldots,h_d)$. The stronger $s$-Cohen–Macaulay $M$-vector conjecture. The vector

$$(h_0,\ h_1-h_0,\ \ldots,\ h_{\lceil\frac{d+s}{2}\rceil}-h_{\lceil\frac{d+s}{2}\rceil-1})$$

is an $M$-vector. This would strengthen the usual $g$-theorem-type conclusion for Cohen–Macaulay complexes, but the approach described in the paper gives no clue how to prove the asserted injection or this formulation.